1

ELASTIC-PLASTIC MODE-II FRACTURE

OF ADHESIVE JOINTS

Q. D. Yang1, M. D. Thouless1,2 and S. M. Ward3

1Department of Mechanical Engineering & Applied Mechanics2Department of Materials Science & EngineeringUniversity of Michigan, Ann Arbor, MI 48109

3Scientific Research LaboratoriesFord Motor Company Dearborn, MI 48121

Abstract

A numerical study of the elastic-plastic mode-II fracture of adhesive joints is

presented in this paper. A traction-separation law was used to simulate the mode-II

interfacial fracture of adhesively-bonded end-notched flexure (E.N.F.) specimens loaded in

three-point bending, with extensive plastic deformation accompanying failure. The

fracture parameters for the traction-separation law were determined by comparing the

numerical and experimental results for one particular geometry. These parameters were

then used without further modification to simulate the fracture of other E.N.F. specimens

with different geometries. It was found that the numerical predictions for the loads and

deformation were in excellent agreement with the corresponding experimental results.

Keywords: adhesive joints, fracture, plasticity, numerical modeling, mechanical testing

(Revised: May 2000)

2

1. Introduction

Joining two materials by a third phase is of long-standing interest in a variety of

industrial and technological applications, including traditional adhesive bonding, brazing

and soldering, and composite materials. For elastic adhesive joints, in which the

deformation of the adherends is dominated by elasticity during the fracture process, the

conditions for fracture can be predicted using linear-elastic fracture mechanics (L.E.F.M).

A substantial amount of work has been done in this area, and a detailed review of the

appropriate mechanics has been given by Hutchinson and Suo (1992). However, the

development of suitable analyses for the fracture of adhesive joints where the adherends

deform plastically during the fracture process still remains incomplete because of the

complications caused by the coupling between macroscopic plasticity in the adherends

and the actual fracture process.

It has been demonstrated that the use of a traction-separation relation to

characterize the fracture process is a very promising technique to deal with coupled

plasticity and fracture (Hutchinson and Evans, 1999). A number of groups (Needleman,

1987; 1997, Tvergaard and Hutchinson, 1992; 1993; 1994, Wei and Hutchinson, 1997;

1998) have demonstrated that toughness is not the only controlling parameter when

plasticity accompanies fracture: the cohesive stresses exerted at the crack-tip also play a

very important role. Fracture can be simulated by using an embedded-process-zone (or

cohesive-zone) model in which the toughness and peak cohesive stress are used as the

two major parameters characterizing the fracture process. By incorporating such a model

3

into nonlinear finite-element analyses which account for plasticity in the surrounding

materials, the energy dissipated by plasticity can be separated from that absorbed during

the decohesion process along the crack plane.

The present work is motivated by the authors’ attempt to establish a general

modeling approach to obtain quantitative predictions for the fracture of adhesive joints

that are deforming plastically. The core of this approach is to use a cohesive-zone model

to mimic the role that the adhesive layer plays during the deformation and fracture of the

joints. The interfacial toughness and cohesive stresses are incorporated in the traction-

separation law of cohesive-zone elements that interact with the surrounding material. By

doing this, the energy associated with fracture is completely separated out from the

energy dissipated by gross plasticity in adherends. This approach has been successfully

used to predict the mode-I fracture of various adhesive joints failing with extensive plastic

deformation (Yang et al., 1999; 2000). An experimental procedure for determining the

parameters of a suitable cohesive-zone model was proposed in these studies, and it was

shown that numerical predictions based on these parameters were in excellent agreement

with experimental results. While these studies demonstrated the success of the approach

for mode-I fracture, establishing a general framework requires an understanding of mixed-

mode fracture. As a first step towards this goal, a study of pure mode-II fracture is

described here. An extension to the general mixed-mode case, based on this work and the

earlier mode-I studies, has recently been developed (Yang and Thouless, 2000).

4

Three-point bending of adhesively-bonded, end-notched flexure (E.N.F.)

specimens (Fig. 1) was chosen to provide the mode-II geometry. Stable crack growth

occurs before the crack reaches the loading point in this type of specimen, and it provides

an ideal case study for mode-II fracture. Furthermore, the test has previously been used

to evaluate the mode-II fracture toughness of elastic adhesive joints based on L.E.F.M.

concepts (Barret and Foschi, 1977; Chai, 1988). Chiang and Chai (1994) studied the

plasticity in the adhesive layer near the static crack-tip of elastic E.N.F adhesive joints

using large-strain finite-element analyses. The deformation of the adhesive layer in the

crack-tip region is essentially dominated by shear (with some compressive normal

stresses at the crack-tip). However, only elastically-deforming adherends were

considered in these earlier studies. When plasticity occurs, additional complications arise

not only from the coupling effects between the fracture process and plasticity in

adherends, but also from the geometric non-linearities caused by large deformations and

rotations of the specimens. Therefore, a full nonlinear numerical analysis incorporating an

appropriate cohesive-zone model is necessary for modeling the fracture of such joints.

In this paper, a mode-II fracture problem is analyzed by adapting the technique

previously used with success to model mode-I fracture (Yang et. al., 1999). As shown in

Fig. 2, the shear traction-separation relationship of a mode-II cohesive zone is

characterized by ΓIIo, the work of separation per unit area of crack growth (equal to the

area under the curve), the peak shear stress supported by the bonding tractions, ˆ , and

two shape parameters δt1/δtc and δt2/δtc. Numerical experiments confirmed that ΓIIo and ˆ

5

are the two key parameters that dominate the numerical simulations; the two shape

parameters are less important. Therefore, in the work that follows, the two key

parameters are first determined and then used to predict fracture. Comparisons between

the numerical predictions and the experimental observations of crack propagation,

deformation and fracture are used to validate the numerical technique.

2 Experiments

2.1 Torsion test of adhesively-bonded butt joints

Torsion tests of adhesively-bonded butt joints were conducted to obtain the shear

properties of an adhesive layer (XD4600, from Ciba Specialty Chemicals) with a bond-

line thickness of 0.25 mm. This is identical to the thickness used subsequently for the

E.N.F. specimens. This is essential because it is expected that the shear properties of

adhesive layers depend upon the bond-line thickness (Chai, 1993). The butt joint was

made of two solid aluminum cylindrical rods with a diameter of 12.7 mm (see the inset of

Fig. 3a). The two rods were bonded together by the adhesive and carefully aligned by

clamping them to a V-box during curing. The bond-line thickness was controlled by using

uniform-sized silica spheres with a diameter of 0.25 mm as spacers. Subsequent

measurements confirmed that the actual thickness was indeed 0.25 mm after the

specimens were cured.

The shear test was performed on a torsion testing machine with a loading rate of

0.05 rad/min. The gauge length of the sample was 98 mm. In addition, an aluminum rod

cut from the same batch and with the same gauge length was tested independently so that

6

the elastic strain in the aluminum rods could be eliminated from the butt-joint test data.

Curves of applied torque against rotation were recorded. After subtraction of the elastic

rotation in the aluminum rods, a curve of applied torque versus relative rotation angle

across the adhesive layer was obtained. This is shown as a solid line in Fig. 3(a). A

numerical analysis using ABAQUS (5.8) with axisymmetric elements was performed to

back out the shear stress-strain curve of the adhesive. The resultant curve is shown in

Fig.3(b). The results of the numerical simulation of the torsion test based upon this

constitutive relation are shown as a dashed line in Fig. 3(a). The shear stress-strain curve

for the adhesive given in Fig. 3(b) provided guidance in the subsequent determination of

the mode-II fracture parameters used in the cohesive-zone model.

2.2 Three-point bending test of adhesively-bonded E.N.F. specimens

Edge-notched flexure specimens were fabricated using 20 mm wide coupons cut

from aluminum sheets (5754 alloy from Alcan Rolled Products Company) of different

thicknesses varying between 1.6 mm and 3.0 mm. The true stress-strain curve of the

aluminum, which is essential for the numerical simulation to be introduced in next section,

was obtained from standard uniaxial tensile tests and is shown in Fig. 4. The coupons

were bonded by a layer of the commercial adhesive XD4600. The adhesive layer was

kept at a uniform thickness of 0.25 mm using silica spheres as spacers. An initial crack

along one of the adhesive/aluminum interfaces was introduced by inserting a strip of

Teflon® tape into the interface before applying the adhesive. After curing, any excess

adhesive was removed by careful filing and polishing of the specimen edges. To facilitate

7

experimental observations, straight lines were cut vertically across the bond-lines using a

sharp razor.

The specimens were placed over a test fixture consisting of two supporting

cylinders. A third cylinder connected to the load cell of a tensile machine was used to

bend the sample at its mid-point (Fig. 1). The span between the two supporting

cylinders was 60 mm, and the initial crack length was 20 mm. To reduce possible

frictional effects along the cracked surfaces, a glass fiber with a diameter of 0.4 mm was

inserted as a spacer between the adherends at the crack mouth. The force and relative

displacement between the center loading point and the supports were recorded as the

specimens were deformed. The deflection measurements were made using a high-

resolution CCD camera. This camera was also used to monitor the crack propagation and

the deformation of the adhesive layer near the crack-tip. A series of tests were performed

for each thickness of aluminum. Each test was performed at room temperature, with a

cross-head velocity of 2 mm/min.1

For all the specimens tested in this paper, the fracture was purely interfacial and

the crack propagated along the interface between the adhesive layer and the compressive

adherend.2 The deformation of a 2.3 mm thick specimen immediately after the crack

began to grow is shown in Fig. 5(a), and a magnified view of the crack-tip region is shown

in Fig. 5(b). Figure 5(a) indicates the extent of frictional sliding between the sample and

1 Experimental results show that there is no discernable difference between the load-displacement curves ofsamples tested at displacement rates of 0.2mm/min and 200mm/min.

8

the supports. The lines at the ends of the sample indicate the initial location of the

contact points between the supports and the sample. The scribe marks in Fig. 5(b) show

that the nominal shear strain across the bond-line is about 40% at a distance

approximately twice the adhesive thickness ahead of the crack-tip. An even larger local

strain is expected in the immediate vicinity of the crack-tip, as also observed by Chai

(1992). The full extent of the plastic zone, measured from the crack-tip to the farthest

point ahead of the tip which shows visible straining, was about 20 times the bond-line

thickness.

3. Numerical simulations and comparisons to experimental results

Numerical calculations were performed to simulate the three-point bending tests.

These calculations incorporated a cohesive-zone model within elastic-plastic finite-

element calculations using the ABAQUS code (version 5.8). Except for the contact

regions between the glass fiber and the adherends, where three-point elements were used

in order to obtain a finer mesh, four-point bi-linear elements were used for the adherends.

Reduced integration was used for all these elements in order to prevent possible shear-

locking effects. Large-strain and large-rotation conditions were considered, with the von

Mises yield criterion and an isotropic strain-hardening model being used to simulate the

mechanical behavior of the adherends. Two-dimensional analyses were used for the sake

of numerical efficiency. It should be noted that plane-stress elements provided a better

approximation to this geometry than plane-strain elements. Since this is not an obvious

2 The crack always kinked into the interface immediately after it started to grow even if it was initiated atthe mid-thickness of the adhesive layer.

9

point, a more detailed discussion is given in the Appendix. Another important feature of

this study was the use of contact elements to simulate the sliding between the supporting

cylinders and the adherends, and between the glass fiber and the adherends. As can be

seen from Fig. 5(a), there was fairly extensive sliding between the supporting cylinders

and the adherends; this resulted in a decrease in the effective span length as the specimens

deformed. Numerical experiments showed that failure to use contact elements between

the supporting cylinders and adherends could result in as much as a 15% reduction in the

predicted load. Therefore, it was deemed essential to use contact elements to allow the

modeling of the sliding between the supporting cylinders and adherends, and between the

glass fiber (spacer) and adherends.

In the numerical calculations, the adhesive layer was replaced by user-defined

elements that simulated the appropriate traction-separation relationship of a mode-II

cohesive-zone model. As described earlier, the work of separation per unit area, ΓIIo, and

the peak shear stress supported by the interface, ˆ , are the most important parameters

that need to be determined for such a model. Systematic numerical experiments confirmed

that the two shape parameters are less important, and they were chosen to be constants of

δt1/δtc=0.15 and δt2/δtc=0.5 for all the simulations. The critical shear displacement, δtc, is

fixed by the choice of the shape parameters, ΓIIo and ˆ , since ΓIIo = 0.5 ˆ (δtc - δt1 + δt2).

The three-point bending tests showed that the local strain in crack-tip region was

at least 40%. Therefore, it is evident from Fig. 3(b) that the peak shear stress can be

10

taken as ˆ = 35 MPa. The mode-II joint toughness, ΓIIo was then determined to be

5.4 kJ/m2 by using different values of ΓIIo to predict load-deflection curves for the 2.3 mm

specimen, and finding the best fit to the associated experimental results. The resultant

curve using these two parameters is shown in Fig. 6 together with a typical experimental

curve. The oscillation in the numerical curve is caused by the difficult convergence

procedure for the contact elements between the glass fiber and the adherends. The

maximum loads in both the experimental and numerical curves correspond to the points at

which the crack began to grow. The crack was observed to propagate steadily as the

displacement was increased, but under a gradually decreasing load. It can be seen that the

numerical calculation does an excellent job of reproducing the entire deformation history

both before and after the beginning of crack growth. As a contrast to the numerical

calculations using cohesive-zone elements, the results of a finite-element calculation using

the continuum properties of the adhesive with no failure criterion are superimposed on

Fig. 6. While such a calculation captures the deformation behavior until the crack begins

to grow, the absence of a failure criterion renders it invalid after this point. However, the

comparison shows that, in this particular case, the cohesive tractions are basically

mimicking the deformations of the adhesive layer.

After being validated using the 2.3 mm-thick samples, these values of

ˆ = 35 MPa and ΓIIo = 5.4 kJm-2 were then used without any further modification to

simulate the fracture of other specimens with different thicknesses of the aluminum. The

numerical predictions for the applied loads and deflections are shown as solid lines in

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Fig. 7. The shaded areas in this figure indicate the ranges of the corresponding

experimental data. The numerical results are in excellent agreement with the experimental

results; they not only reproduce the magnitudes of the forces, but also capture some of

the nuances of the shapes of the load-deflection curves. However, since plane-stress

conditions were assumed in all the simulations, it is not surprising to see that the

numerical results are closer to the lower bounds of the experimental data.

The numerical prediction for the global deformation of a 2.3 mm thick specimen at

the point when the crack begins to grow is shown in Fig. 8(a). A magnified view of the

deformation and shear stress distribution near the crack-tip is shown in Fig. 8(b). A

comparison between these numerical predictions and the experimentally-observed

deformation (Fig. 5a) shows excellent agreement between the two. The predicted size of

the plastic zone can be inferred from the extent of the shear deformation shown in

Fig. 8(b); at approximately 20 times the bond-line thickness. This again agrees with the

experimental observations (Fig. 5b).

4. Discussion

The pure mode-II interfacial toughness of the joint (ΓIIo = 5.4 kJm-2) is much

higher than the associated interfacial mode-I value of ΓIo = 1.0 kJm-2 (with a peak normal

cohesive stress ̂ = 60 MPa), which was determined from wedge-induced fracture tests

of double-cantilever beams bonded by an identical thickness of the adhesive (Yang and

12

Thouless, 2000).3 This demonstrates that the fracture is highly mode-dependent. Similar

results were also observed by Chai in his systematic studies of the fracture of elastic

adhesive joints (Chai, 1988; 1992). These studies showed that, except for extremely thin

adhesive layers (<10 µm), the typical value of ΓIIo was about an order of magnitude higher

than the corresponding ΓIo value. A major contribution to the high value of ΓIIo is the

higher capability of the adhesive layer to deform plastically in response to shear. In the

interfacial mode-I fracture tests, the maximum tensile strain across the adhesive layer

before fracture was observed to be only about 8%. The maximum shear strain observed in

the torsion tests of the butt joints was about 60% (Fig. 3b). Therefore, much more

energy is dissipated in the adhesive layer under mode-II loading than under mode-I

loading. A rough estimate for the values of the mode-I and mode-II toughnesses can be

obtained from the product of the critical displacements and cohesive stresses. For a

0.25 mm thick adhesive layer, the data given above would indicate a mode-I toughness of

about 1 kJm-2 and a mode-II toughness of about 5 kJm-2. The large difference in the

cohesive-zone parameters for mode-I fracture and mode-II fracture emphasizes the need

for mode-dependent traction-separation relations to analyze mixed-mode fracture. This

work has now been completed and will appear elsewhere (Yang and Thouless, 2000).

5. Conclusions

A pure mode-II cohesive-zone model has been used to simulate the elastic-plastic

fracture of adhesively-bonded, end-notched flexure specimens loaded by three-point

3 The value for ΓIo quoted here is for interfacial failure. A value of ΓIo =1.4 kJ/m2 and ̂ = 100 MPa was

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bending. The fracture parameters of the model were determined by matching the

numerical results with the associated experimental results. These parameters were then

used without any further modifications to predict the fracture of other adhesive joints

with different thicknesses of adherends. The numerical predictions were in excellent

agreement with the associated experimental results. Therefore, it has been demonstrated

that it is possible to predict the loads associated with crack growth, and to model the

entire deformation history of plastically-deforming adhesive joints loaded in shear.

Acknowledgments

This work was supported by NSF Grant CMS-9624452 and Ford Motor

Company. The authors would also like to express their appreciation to Prof. A. Waas and

Mr. C. S. Yerramalli in the Dept. of Aerospace Engineering of University of Michigan for

their help in the torsion test.

found for mode-I fracture within the adhesive layer (Yang et al., 1999).

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References

ABAQUS Finite Element Analysis Code and Technical Manual, Version 5.8, Hibbit,

Karlsson and Sorensen Inc., 1998

Barret, J. D. and Foschi, R. O., Engineering Fracture Mechanics, 9 (1977) 371-378

Chai, H., International Journal of Fracture, 37 (1988), 137-157.

Chai, H., International Journal of Fracture, 58 (1992), 223-239.

Chai, H., Journal of Materials Science, 28 (1993), 4944-4956

Chiang, M. Y. M. and Chai, H., International Journal of Solids and Structures, 31 (1994),

2477-2490.

Hutchinson, J. W. and Evans, A. G., Harvard University, Division of Engineering &

Applied Sciences, Report MECH-351, 1999.

Hutchinson, J. W. and Suo Z., Advances in Applied Mechanics, 29 , (1992), 63-191

Needleman, A., Journal of Applied Mechanics, 54 (1987), 525-531.

Needleman, A. Computational Mechanics, 19 (1997), 463-469.

Tvergaard, V. and Hutchinson, J. W., Journal of the Mechanics and Physics of Solids, 40

(1992), 1377-1397.

Tvergaard, V. and Hutchinson, J. W., Journal of the Mechanics and Physics of Solids, 41

(1993), 1119-1135.

Tvergaard, V. and Hutchinson, J. W., Philosophical Magazine A, 70 (1994), 641-656.

Wei Y. and Hutchinson, J. W., Journal of the Mechanics and Physics of Solids, 45

(1997), 1137-1159.

Wei Y. and Hutchinson, J. W., International Journal of Fracture, 93 (1998), 315-333.

15

Yang, Q. D., Thouless, M. D. and Ward, S. M., Journal of the Mechanics and Physics of

Solids, 47 (1999), 1337-1353.

Yang, Q. D., Thouless, M. D. and Ward, S. M., Journal of Adhesion, 72 (2000), 115-132.

Yang, Q. D., and Thouless, M. D., submitted to International Journal of Fracture,

(2000).

16

Appendix

While a full 3-D calculation is required for a rigorous analysis of this problem, a

2-D approximation was used in the interests of numerical efficiency. Plane-stress

elements were used in these calculations despite the fact that, at first glance, the samples

might appear to be under plane-strain conditions. The reason that plane-stress elements

proved to be more useful in this particular study is that the adhesive layer is much more

compliant than the aluminum adherends, and does not provide the constraint necessary

for plane-strain deformation of the specimen. To demonstrate this point, a three-point

bend test was performed on an adhesive joint made from 2.0 mm thick aluminum coupons

with no pre-crack. Figure A1 shows the deformation in both the adherends and the

adhesive. The relative displacements between the upper and lower beams, which result

from shear deformation in the adhesive layer, are clearly seen in this figure. These

displacements allow the two beams to deform in a relatively “independent” fashion--

different from the predictions of simple beam theory. The stiffness of the composite

beam is much lower than that of a solid beam with the same thickness.

This point is further demonstrated by the force versus deflection curves shown in

Fig. A2. In this figure, the experimental results are shown by the solid line. Numerical

results of 2-D plane-stress, 2-D plane-strain and 3-D finite-element analyses using a

continuum model for the adhesive (not a cohesive-zone model) are also shown in this

figure. It is evident that the 2-D plane-stress and 3-D4 results were in agreement with the

4 In this analysis, a relatively coarse mesh was used due to the limited memory size of the computers.Therefore, the 3-D results should be regarded as the upper limit of the actual load-displacement curve.

17

experimental data, while the 2-D plane-strain assumption considerably over-estimated the

beam stiffness.

18

Figure Captions

Figure 1 Test configuration of an end-notched flexure (E.N.F.) specimen. The span,

2L, is 60 mm, and the initial crack length, a, is 20 mm.

Figure 2 The traction-separation law used in the numerical analyses. It is

characterized by a work of separation per unit area, ΓIIo, a peak shear

stress, ˆ , and two shape parameters δt1/δtc and δt2/δtc.

Figure 3 (a) Experimental curve of applied torque versus relative rotation across the

adhesive layer of butt joints (solid line). Also shown in this figure is the

numerical results predicted by the constitutive properties shown in

Fig. 3(b) (dashed line).

(b) Constitutive properties of the adhesive XD4600 backed out from the

torsion test of adhesively-bonded butt joints.

Figure 4 The true stress-strain relation of the adherend material (5754 aluminum

alloy).

Figure 5 (a) Micrograph of a deformed 2.3 mm thick edge-notched flexure sample at

the point at which the crack has just begun to grow.

(b) A magnified view of the deformations in the vicinity of the crack-tip.

Figure 6 Load (per unit width) for a 2.3 mm thick edge-notched flexure specimen

plotted as a function of the deflection at the mid-point of span. Three

19

curves are compared: (i) experimental data, (ii) numerical results of a

cohesive-zone model with ˆ = 35 MPa and ΓIIo = 5.4 kJm-2, and (iii)

numerical results using a continuum model for the adhesive layer with no

failure criterion.

Figure 7 Load (per unit width) plotted as a function of the deflection at the mid-

point of the span for 1.6, 2.0, 2.3 and 3.0 mm thick, edge-notched flexure

specimens. The shaded areas show the range of experimental results. The

solid lines show the results of numerical calculations with ˆ = 35 MPa

and ΓIIo = 5.4 kJm-2.

Figure 8 (a) Numerical predictions for the deformation and the von Mises stress

distribution in a 2.3mm edge-notched flexure sample at the point at which

the crack has just begun to grow.

(b) Numerical predictions for the distribution of shear stresses and for the

shear displacements across the interface.

Figure A1 The relative deformation across the adhesive layer in an adhesively-bonded

beam made with 2.0mm thick aluminum adherends and tested in three-

point bending. The span was 60 mm between the outside supports. (The

shear displacements disappeared at the central loading point to the right of

the figure.)

20

Figure A2 Numerically-predicted and experimentally-observed results for the load

(per unit width) needed to deform an adhesively-bonded beam made with

2.0mm thick aluminum adherends and tested in three-point bending (with a

span of 60 mm between the outside supports). The numerical results were

obtained using 2-D plane-stress, 2-D plane-strain and 3-D finite-element

elements with a continuum model for the adhesive.

L L

a

Fig. 1

Fig. 2

IIoΓ

t2t1 tc t

ˆ

0

5

10

15

20

0 0.005 0.01 0.015 0.02 0.025

App

lied

torq

ue,

N*m

Relative rotation across the adhesive layer, radius

Experimental

Numerical

98 mm

12.7 mm

0.25 mm

Fig. 3(a)

0

5

10

15

20

25

30

35

40

0 0.1 0.2 0.3 0.4 0.5 0.6

Shea

r st

ress

, M

Pa

Shear strain

Fig. 3(b)

0

50

100

150

200

250

300

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16

Tru

e st

ress

, MPa

True strain

Fig. 4

Fig 5 (a)

5 mm

Fig. 5(b)

Crack-tip

Plastic zone

1 mm

0

10

20

30

40

50

60

0 1 2 3 4 5

load

per

uni

t wid

th, N

/mm

Deflection at mid-point of span, mm

Numerical results using continuumelements for the adhesive layer

Numerical results using EPZ cohesiveelements for the adhesive layer

Experimental results

Fig. 6

0

20

40

60

80

100

0 1 2 3 4 5 6 7 8

Loa

d pe

r un

it w

idth

, N

/mm

Deflection at mid-point of span, mm

h=1.6 mm

h=3.1 mm

h=2.0 mm

h=2.3 mm

Fig. 7

Fig. 8(a)

Crack-tip

5 mm

Fig. 8(b)

1 mm

Fig. A1

1 mm

0

10

20

30

40

50

60

70

0 1 2 3 4 5

Forc

e pe

r un

it w

idth

, N

/mm

Deflection at mid-point of span, mm

Numerical result(2-D plane strain)

Numerical result (3-D)

Numerical result(2-D plane stress)

Experimental result

Fig. A2